In a class of 36 students, where it can be, shall we say, difficult for me to do formative assessment on every student every day, the Talking Points structure give me a great way to surface and deal with student misconceptions by getting students to surface, discuss, and correct them.
Our sequencing for Geometry has gotten totally screwy this year because of some new district requirements around the Common Core. Having learned how to do all of the basic constructions, we are now finally approaching the unit test on parallel lines and their angles. There are so many possible errors in understanding that can happen around these, I wanted to create a group work activity to address them. I also want to change groups up this week, so I am using this as community-building as well.
I've discovered it is a good idea to weave together community-building statements, growth mindset statements, metacognitive self-monitoring statements, and Always / Sometimes / Never statements. The Talking Points structure lets me accomplish multiple goals simultaneously, which is something I need to do with such big classes.
Einstein was right — imagination IS more important than knowledge.
I used to think of that quote a lot as I walked the Princeton campus, through quadrangles and past trees that — legend had it — he had crashed into on his bicycle while he was lost in his thoughts.
I passed his house on Mercer Street every week on my way to music lessons, and I wondered if his sister had not had the door painted it bright red so that he might notice it and not bump into it.
When I realized I would have to teach both constructions AND proof this year in Geometry, I first thought of them as a Big Obstacle. But seven weeks in and I have fallen in love with constructions. I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.
I've been looking for a way to do formative assessment that also doubles as a practice activity, and I seem to have stumbled on something good with my new "Lightning Rounds Review With Whiteboards activity. It uses some of what I appreciate most from both Steve Leinwand and Dylan Wiliam.
We have large-ish whiteboards — enough for one per table group — in our classrooms. Since I've also got around 36 students in each of my classes, I've been drowning with my older methods of checking for understanding. With 175 total students, even a 1-minute-per-exit-ticket assessment can take ages to go through — plus all the time it takes me to make it up.
This activity has been working much better. It also takes less prep up front and allows me to engage more deeply with the students and/or groups who need more attention. Assessing 9 tables' work at a glance is much more manageable and less exhausting than trying to flip through 36 pieces of student work. And of course, that is better for everybody.
Here is how it worked today.
In Precalculus, we're working on evaluating trig functions of real numbers on the unit circle. Students and groups are strongly encouraged to use their unit circles and notes to help them gain confidence and fluency.
Each table group gets a board, three markers, and an eraser. On the document camera, covering the non-current rows with a folder, I reveal one row of problems, which contains three problems: (a), (b), and (c).
Students discuss and work the problems, and when they are done, they hold up their table's whiteboard for a "check-in."
I glance around the room and check three answers at a time, calling out, "Table 2, you are checked in and correct!" "Table 6, you are checked in and correct!" "Table 1, you need to reexamine part (b)."
Because there are three problems in each "round," there's plenty of time for groups to make an error, assess their work and reconsider, and re-present their findings. Because there are only nine tables reporting in, it is manageable from both a teacher and a student perspective.
Plus, of course, since there is a group whiteboard and markers to play with, those groups who finished quickly and received their checkpoint are happy to doodle or play tic-tac-toe or offer funny editorial comments or cartoons while other groups receive some attention or extra time.
Last night was our Back To School Night, and student clubs were selling food as fundraisers for their clubs. Two of my students discovered that my classes' parents and I were helpless in the face of their delicious goodies. At one point, I had told them, Look, I don't have cash; my purse is locked upstairs in my desk.
Around the 6th or 7th round, Table 7 included an editorial comment on the side of their board: "We heart Dr. S! Also, you owe us $2. :)" While everybody started working the next round of problems, I unlocked my purse out of my closet and paid my debts. Some good laughs were had by all, and it was a nice piece of mathematical community-building.
I am still drowning in papers to grade, but I wanted to write a quick post before the week got too far away from me. I had my first-ever bomb threat this past week, and it rattled us all. I grabbed my iPhone, my MacBook Pro, and my math pencil and escorted my class out of the building.
I had to make my way home with no keys, no purse, and no money. Just carrying my iPhone and my laptop and my trusty math pencil.
As I battled public transit to make my way home (with no money – just explaining where I'd been and what had happened), it occurred to me that everybody was going to be nuts the next day. So I started writing some new Talking Points for us to do about the events of the previous day.
One of the things that becomes clear with training and time is that as a teacher, my job is to provide an emotional "container" for the experiences we have in my classroom. I decided to use Talking Points as a way of enabling students to process their experience within the structure we've been building for several weeks now.
My students' responses and respect for the structure blew my mind. And they made it possible for us to lose only one day rather than several. And by Thursday, our classroom community had returned to a steady enough state to move ahead.
I'm required to teach two-column proofs in Geometry, but having also been trained as an English teacher, this has never seemed like a problem to me. In fact, if anything, the activity I used on Friday seemed to scaffold the process using students' existing knowledge better than anything else I've tried before.
My design process was as follows: because of their ELA and writing backgrounds, students already know far more about constructing an argument in words and statements than we math teachers often give them credit for knowing. All the major writing curricula, such as Jane Schaffer and Six Traits, provide scaffolded methods for teaching students to make claims and to support their assertions with evidence and interpretations that connect that evidence to their claims through interpretive statements. Indeed, the Jane Schaffer method, in particular, has a very lovely scaffolded process (which I've extended in the past) to bridge students' metacognitive processes about their writing, taking them from a place of very concrete thinking to one of considerable abstraction.
So why not use this same kind of process for proof?
Instead of having students merely "fill in" the reasons for the statements in their first proof (in our curriculum, that's the Midpoint Theorem), I created a task card with instructions and materials for creating a "working poster" (an idea I have adopted from Malcolm Swan) of a two-column proof. They needed to set it up the way we'd done it the day before (two columns, Statements and Reasons), and then they would need to (a) sort their cut-out statements from the task card into a correct order (more than one order is possible), and (b) use their notes and discussions to give the justification or reason that permitted them to make each of these assertions in turn.
The richness of their conversations blew me away. They also confirmed my intuitions that (1) math conversations and projects can indeed draw on students' existing competencies in argumentation that they have developed in their English and Social Studies classes (indeed, many relished the opportunity!), and (b) it is indeed possible to create intellectual need (see Guershon Harel and Dan Meyer) for definitions, postulates, previous theorems, and propositions from algebra through situational motivation.
This activity turned two-column proof into a reasoning and sense-making activity that exposed and built on prior knowledge instead of invalidating it; created what Swan calls "realistic obstacles to be overcome"; turned students' notes into a valued and valuable learning resource; and used higher-order questioning, as opposed to mere recall.
I realize I have not referenced the van Hiele levels here, but that is, in part, because I think I may be kind of bypassing some of their assumptions. I'm not at all sure about this, though, and I would welcome better-informed thoughts and thinking about this in the comments.
My Geometers took the opportunity to inform me through their Chapter 1 exams that they really don't get how angles are named. So this seemed like a perfect opportunity for more Talking Points, of course. :)
This time I'm giving everybody a diagram of a figure that the Talking Points refer to. They will have to do some reasoning about naming angles in order to do the Talking Points. They love doing Talking Points, but they mostly like coming to immediate consensus. Hopefully this will throw a monkey wrench (so to speak) into those works.